Problem: Solve the equation. $\dfrac{dy}{dx}=\dfrac{x^2}{6e^y}$ Choose 1 answer: Choose 1 answer: (Choice A) A $y=\pm\sqrt{-2e^{-x}+C}+10$ (Choice B) B $y=\pm\sqrt{-2e^{-x}}+C$ (Choice C) C $y=\ln\left(\dfrac{x^3}{18}+C\right)$ (Choice D) D $y=\ln\left(\dfrac{x^3}{18}\right)+C$
We can bring this equation to the form $f(y)\,dy=g(x)\,dx$ : $\begin{aligned} \dfrac{dy}{dx}&=\dfrac{x^2}{6e^y} \\\\ 6e^y\,dy&=x^2\,dx \end{aligned}$ This means we can solve this equation using separation of variables! $\begin{aligned} 6e^y\,dy&=x^2\,dx \\\\ \int 6e^y\,dy&=\int x^2\,dx \\\\ 6e^y&=\dfrac{x^3}{3}+C_1 \\\\ e^y&=\dfrac{x^3}{18}+C \\\\ \ln(e^y)&=\ln\left(\dfrac{x^3}{18}+C\right) \\\\ y&=\ln\left(\dfrac{x^3}{18}+C\right) \end{aligned}$ [Where did we get C?] Notice that after the integration, more work was required in order to isolate $y$. In conclusion, this is the solution of the equation: $y=\ln\left(\dfrac{x^3}{18}+C\right)$